I think that this presentation of the numbers is excellent in form and fluency, but he could have left a little more time in the first time presentation for beginners' repetition. For those "reading along" with teacher for review, however, the speed - word speed and spacing - is really good.

I believe that he is correct in teaching the neuter form of 1, 3 and 4 as the one used for counting. I want to discuss the choice between μηδέν and οὐδέν for "zero". As might be expected by bringing it up for discussion, I am in fact saying that I doubts, misgivings or perhaps disagreement with his choice of οὐδέν. Since N.J. Erickson himself is unlikely to be part of this discussion, I'll be a bit less goading, and won't play my cards as close to my chest as I am want to.

A numeral is an adjective when used together with a noun - "two apples" (δύο μῆλα) - and a substantive when is exists by itself in an abstracted sequence of counting - "the number 'two'". As stated, in his video, N.J. Erickson uses the neuters ἕν, τρία, and τέσσαρα in his counting sequence. I think that is theoretically correct according to the way Greek uses the article with a neuter adjective to express the (abstracted) "idea" of something - a number in a sequence being an abstraction of successive numbers role as the identifiers of an ever increasing quantity of something - the abstract identification of quantity.

Now, on to the difference between μηδέν and οὐδέν:
If, as it seems from the greater part of the examples, that the factor most strongly influencing the pattern of this bivalent (μηδέν and οὐδέν) distribution in most cases is syntactic, then it is in the marginal distribution pattern that the non-syntactically determined patterning will be found. Here are two examples, which I believe are from the margins.

The "nothing" (οὐδὲν) is something that exists. It is part of the subset of the "all" (πᾶν). When we think of nothing, we can think of the things, so nothing (οὐδὲν) is actually "not (even) one of the things". Because the nothing is a something, is a something in this example, then it could be suggested that when οὐδέν is used then it might be that the speaker is suggesting the listener think of a few examples.

It is a blanket "zero", no mathematical figure (μηδέν) more (πλέον) than prescribed. Μηδέν, which does not seem to connotate or suggest a concrete exam, has a "don't even think about it" sense. Look at a few examples that might be thought of falling somewhere nearer to the margins of the distribution:

Ὅρα, μηδενὶ μηδὲν εἴπῃς· (Mark 1:44), the the nothing and the nobody are completely nothing and nobody. Εὔχομαι δὲ πρὸς τὸν θεόν, μὴ ποιῆσαι ὑμᾶς κακὸν μηδέν, (2 Corinthians 13:7), Paul is not prompting them to think about evil things and then NOT do them. ἵνα ὁ ἐξ ἐναντίας ἐντραπῇ, μηδὲν ἔχω περὶ ἡμῶν λέγειν φαῦλον. (Titus 2:8), Paul is giving advice about how there can be nothing (really nothing that a person who is really looking for something), that can be said.

In consideration of the extent of context, we could think of a simple example. "I have no apple.", is complete in itself, and it means completely not have, which I would theoretically possit as a μηδέν. "Mum gave him three, but nothing to me.", where what is lacking is envisaged, I would theorise that as an οὐδέν.

Another way to say that is that it may be that οὐδέν refers to what is, and μηδέν tends to refer to what actually doesn't exist. There is scope for overlap in sutuations where that is required by syntax.

in the abstracted sequence of counting, where nothing is actually nothing, the absence of any number, μηδέν seems more logical. N.J. Erickson's decision to use (or adoption of) οὐδέν doesn't seem logical to me. I think the abstracted value (non-value) of zero is better expressed by μηδέν.

That being all said, let me repeat that I really like N.J. Erickson's youtubes, including this one.

Here I would just think that the choice of μη- has to do with the use of non-indicative forms (the subjunctive in a ἵνα clause and the negation of the imperative). Looking at the examples that you pulled, that's what I see.